Among separable metrizable spaces:

Cantor set is the unique compact zero-dimensional space without isolated points.

$\mathbb Q$ is the unique countable space without isolated points

$\mathbb R \setminus \mathbb Q$ is the unique zero-dimensional, $G_\delta$-space with no compact neighborhood.

$\mathbb Q ^\omega$ is the unique zero- dimensional, first category $F_{\sigma\delta}$-space with the property that no nonempty clopen subset is a $G_{\delta\sigma}$-space.

Question. Is there a simple classification of zero-dimensional $G_{\delta\sigma}$-spaces which have no compact neighborhoods? The simplest examples would be $\mathbb Q$, $\mathbb R\setminus \mathbb Q$, and $\mathbb Q\times (\mathbb R\setminus \mathbb Q)$. Are there many others?

Is there a nice characterization of $\mathbb Q\times (\mathbb R\setminus \mathbb Q)$?


This paper by Van Mill from 1981 gives a characterisation of $\Bbb Q \times \Bbb P$ (where $\Bbb P$ is a common notation for the irrationals) in Thm 5.3:

If $X$ is separable metrisable and zero-dimensional, $\sigma$-complete and nowhere complete and nowhere $\sigma$-compact then $X \simeq \Bbb Q \times \Bbb P$.

Where by complete I mean topologically complete (i.e. in this context: completely metrisable) and a nowhere-$P$ space is one where no non-empty open subset has property $P$, so $\Bbb P$ and $\Bbb Q$ are nowhere locally compact, e.g.)

I think $\Bbb Q \times C$ (with $C$ the Cantor set) is another example for your list.

A lot of information can be found in van Engelen's PhD-thesis from 1985: homogeneous zero-dimensional absolute Borel sets, where he shows there are are $\omega_1$ many homeomorphism types of subsets of $C$ that are both $F_{\sigma \delta}$ and $G_{\delta\sigma}$, also separately written up here. In his thesis he also gave the first characterisation of $\Bbb Q^\omega$ (now relegated to the appendix of it). Look up Van Engelen's work from around that time for related results.


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